Prediction interval in a normal linear model

For the full-rank normal linear model $Y=X\beta +\epsilon$ and an independent future response $Y^{\ast }=x^{\ast T}\beta +{\epsilon }^{\ast }$, putThen an exact $(1-\alpha )$ prediction interval isThe additional one inside the square root is the future observation's irreducible error variance; a confidence interval for its mean omits it.

When a full-rank normal regression with positive residual degrees of freedom happens to fit all supplied observations exactly, its residual estimate of variance is zero. The ordinary Studentized prediction-interval formula formally collapses, but the statistic divides by zero. Under positive true Gaussian variance such samples form a null set; exact rounded or constructed data must not be treated as proof that future randomness vanishes.